On Growth and Form

Citation preview

GROWTH AND FORM

Ah instantdnedus photograph '

'"*'

of a 'splash' of milk.

From Harold

E.

KdgertbTiTirlassachusetts Institute of Technology

See p. 390

ON

GROWTH AND FORM BY

D'ARCY

WENTWORTH THOMPSON

A

new

edition

94

^^

"The

reasonings about the wonderful and intricate operations

of Nature are so

full

of uncertainty, that, as the Wise-man truly

observes, hardly do we guess aright at the things jhat are

and with labour do we find the things that are Stephen Hales, Vegetable Staticks (1727), p. 318, 1738. earth,

upon

before us.''

"Ever since I have been enquiring into the works of Nature have always loved and admired the Simplicity of her Ways." Dr George Martine (a pupil of Boorhaave's), in Medical Essays and Observations, Edinburgh, 1747. I

PREFATORY NOTE THIS

book of mine has

"all preface"

need of preface, for indeed

little

from beginning to end.

have written

I

it is

it

as

an easy introduction to the study of organic Form, by methods which are the common-places of physical science, which are by no means novel

in their application to natural history,

nevertheless naturalists are It

is

but which

accustomed to employ.

little

not the biologist with an inkling of mathematics, but

the skilled and learned mathematician

who must

ultimately deal

with such problems as are sketched and adumbrated here.

have made what use

to no mathematical

skill,

what

have dealt with simple

tools I had;

I

matical methods which simplest kind.

Mr Pope

And

Pope has been mine

cases,

I I

pretend could of

and the mathe-

have introduced are of the easiest and are,

my

book has not been

—the indispensable help^—of many friends.

translating

sought assistance! to

I

Elementary as they

written without the help

Like

I

but

Homer, when

I felt

myself deficient I

the experience which Johnson attributed also,

that

men

of learning did not refuse

to help me.

I

me

wrote this book in wartime, and

during another war.

service

Few

It

was debarred me by

gave

my

are left of the friends

owe them kindly names, that of Dr G. forget the debt I

me

its

employed and occupation, when

revision has

solace

years.

who helped me all.

Let

write

me add

T. Bennett, of

it,

but

I

do not

another to these

Emmanuel

College,

Cambridge; he has never wearied of collaboration with me, and his criticisms

have been an education to

receive.

D. W. T. 1916-1941.

American

edition published

August, 1942

Reprinted

January, 1943

Reprinted

May, 1944

Reprinted

May, 1945

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS PAGE

CHAP. I.

Introductory

II.

On Magnitude

22

The Rate of Growth

78

III.

1

IV.

On the Internal Form and Structure of the Cell

V.

The Forms of Cells

VI.

A Note on

VII.

The Forms of

VIII.

The same

.

286

346

Adsorption

444

Tissues, or Cell-aggregates

.

(continued)

IX.

On

X.

A

XI.

The Equiangular Spiral

XII.

.

Concretions, Spicules, and Spicular Skeletons

.

.

465

.

.

566

.

.

645

Parenthetic Note on Geodetics

741

The Spiral Shells of the Foraminifera

....

748 850

The Shapes of Horns, and of Teeth or Tusks: with a Note on Torsion

874

XIV.

On Leaf-arrangement, or Phyllotaxis

912

XV.

On the Shapes

XIII.

....

of Eggs, and of certain other

Structures

Hollow

....

XVI.

On Form and Mechanical Efficiency

XVII.

On the Theory of Transformations, or the Comparison OF Related Forms

934,

958

1026

Epilogue

1093

Index

1095

Plates

A

Splash of Milk

The Latter Phase of a Splash

Frontispiece " .

.

.

.

.

.

facing page 390

"The mathematicians are well acquainted with the difference between pure science, which has to do only with ideas, and the application of its laws to the use of life, in which they are constrained to submit to the imperfections of matter and the influence of accident." Dr Johnson, in the fourteenth Rambler, May 5, 1750. "Natural History. science."

Sir

.

.is

end of physical The Study of Natvml Philosophy,

either the beginning or the

John Herschel,

in

p. 221, 1831.

"I believe the day must come when the being a mathematician

when he

requires it."

Karl Pearsonf in



without mathematical analysis Nature, January 17, 1901.

—not hesitate to use

biologist will





;

CHAPTER

I

INTRODUCTORY Of

the chemistry of his day and generation,

was a schaft

science,

—for

but not Science

Kant

declared that

that the criterion of true science lay in

mathematics*.

it

eine Wissenschaft, aber nicht Wissen-

This was an old story

:

for

its

relation to

Roger Bacon had

called

and Leonardo da Vinci had said much the samef. Once again, a hundred years after Kant, Du Bois Reymond, profound student of the many sciences on which physiology is based, recalled the old saying, and declared that chemistry would only reach the rank of science, in the high and mathematics porta

strict sense,

when

et clavis

it

scientiarum,

should be found possible to explain che^dcal

reactions in the light of their causal relations to the velocities,

and conditions of equihbrium of the constituent molecules must deal with molecular mechanics by the methods and in the strict language of mathematics, as the astronomy of Newtoii and Laplace dealt with the stars in their courses. We know how great a step was made towards this distant goal as Kant defined it, when van't Hoif laid the firm foundations of a mathematical chemistry, and earned his proud tensions

that, in short, the chemistry of the future

epitaph

Physicam chemiae adiunxitX-

We need not wait for the full reahsation of Kant's

desire, to

to the natural sciences the principle which he laid down.

chemistry

fall

short of

its

nevertheless physiology

apply

Though

ultimate goal in mathematical mechanics §, is

vastly strengthened and enlarged

by

* " Ich behaupte nur dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen konne als darin Mathematik anzutreffen ist" Gesammelte :

Schriften, iv, p. 470.

t "Nessuna humana investigazione si puo dimandare vera scienzia s'essa non passa per le matematiche dimostrazione." X Cf. also

Crum Brown, On an

application of Mathematics to Chemistry, Trans.

R.S.E. XXIV, pp. 691-700, 1867. § Ultimate, for, as Francis Bacon

tells

us:

terminare debet, non generare aut procreare.

Mathesis philosophiam naturalem

INTRODUCTORY

2

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eOor shorter according to species and variety and normal habitat, is a subject now studied under the name of "photoperiodism," and become of great practical importance for the northerly extension of cereal crops in Canada and Russia. Cf. R. G. Whyte and M. A. Oljhovikov, Nature, *

t

Feb.

18, 1939.

§

Kuro Suzuki and T. Hatano, in Proc. Imp. Acad, of Japan, in, pp. 94-96, Withrow and Benedict, in Bull, of Basic Scient. Research, iii, pp. 161-174,

II

Cf. E. C.

I Cf.

1927. 1931.

Teodoresco, Croissance des plantes aux lumieres de diverses longueurs

d'onde, ^WTi. Sc. Nat, Bat.

(8),

pp. 141-336, 1929; N. Pfeiffer, Botan. Gaz. lxxxv,

p. 127, 1929; etc. II See D. T. MacDougal, Influence of light and darkness, etc., Mem. N. Y. Botan. Garden, 1903, 392 pp.; Growth in trees, Carnegie Inst. 1921, 1924, etc.; J. Wiesner, Lichtgenuss der Pflunzen, vn, 322 pp., 1907; Earl S. Johnston, Smithson. Misc.

Contrib. 18 pp., 1938; etc. ness, see

On

the curious effect of short spells of light and dark-

H. Dickson, Proc. E.S. (B), cxv, pp. 115-123, 1938.

THE RATE OF GROWTH

242

Now

if

[ch.

temperature or light affect the rate of growth in

uniformity, aHke in

all

parts

and

strict

in all directions, it will only lead

to local races or varieties differing in size, as the Siberian goldfinch

or bullfinch differs from our own.

But if there be ever

so Httle of a

discriminating tendency such as to enhance the growth of one tissue or one organ

more than another*, then

it

must soon lead

to racial,

or even "specific," difference of form.

hardly to be doubted that chmate has some such dis-

It is

criminating influence.

The large leaves of our northern trees are an instance of it; and we have a better instance of it still in Alpine plants, whose general habit is dwarfed though their floral organs suffer little or no reduction f. Sunhght of itself would seem to be a hindrance rather than a stimulant to growth; fact of a plant turning towards the sun

and the familiar means increased growth on

the shady side, or partial inhibition on the other.

More curious and

still

more obscure

is

the moon's influence on

growth, as on the growth and ripening of the eggs of oysters, seaurchins and crabs. Behef in such lunar influence is as old as Egypt; it is

way

confirmed and in

justified, in certain cases,

which the influence

is

exerted

is

nowadays, but the

unknown J.

quite

Osmotic factors in growth

The curves of growth which we have been studying have a To the morpho-

twofold interest, morphological and physiological. logist,

who has

learned to recognise form as a "function of growth,"

the most important facts are these: orderly phenomenon,

organisms, constants;

each having (2)

(1)

that rate of growth

common

with general features its

own

characteristic

rates,

is

an

to various

or specific

that rate of growth varies with temperature, and so

with season and with climate, and also with various other physical factors, external

and internal to the organism (3) that it varies in and along various directions or axes: ;

different parts of the body, *

Or

as

we might say nowadays, have a

different "threshold value" in

one

organ to another. t Cf. for instance, NageH's classical account of the effect of change of habitat on alpine and other plants, Sitzungsber. Baier. Akad. Wiss. 1865, pp. 228-284. X Cf. Munro Fox, Lunar periodicity in reproduction, Proc. R.S. (B), xcv, pp. 523-550, 1935;

also Silvio Ranzi, Pubblic. Slaz. Zool. Napoli, xi, 1931.

OF OSMOTIC FACTORS IN

Ill]

GROWTH

243

such variations being harmoniously "graded," or related to one another by a "principle of continuity," so giving characteristic

rise

to

the

form and dimensions of the organism and to the

changes of form which

it

To the

phenomenon

physiologist the

considerations,

and

exhibits in the course of

its

development.

of growth suggests

many

other

especially the relation of growth itself to chemical

and physical forces and energies. To be content to shew that a certain

rate of growth occurs in

a certain organism under certain conditions, or to speak of the

phenomenon as a "reaction" of the living organism to its environment or to certain stimuh, would be but an example of that "lack of particularity" with which we are apt to be all too easily satisfied. But in the case of growth we pass some little way beyond these limitations: to this extent, that

an

affinity

with certain types of

chemical and physical reaction has been recognised by a great number of physiologists*.

A

large part of the

more conspicuously to say,

is

phenomenon of growth,

in plants,

is

in animals

and

still

associated with "turgor," that

dependent on osmotic conditions.

is

In other words, the

rate of growth depends (as we have already seen) as much or more on the amount of water taken up into the living cells f, as on the actual amount of chemical metabohsm performed by them; and

sometimes, as in certain insect-larvae, we can even distinguish

between

tissues

which grow by increase of

cell-size,

the result 'of

and others which grow by constituent cells |. Of the chemical phenomena which

multiplication

imbibition,

* Cf.

le

their

result in the

F. F. Blackman, Presidential Address in Botany, Brit. Assoc, Dublin,

The

idea was developpement du

1908.

of

first

enunciated by Baudrimont and St Ange, Recherches sur

foetus,

Mem.

Acad. Sci.

xi, p. 469, 1851.

t Cf. J. Loeb, Untersuckungen zur physiologischen Morphologie der Tiere, 1892; also Experiments on cleavage, Journ. Morphology, vii, p. 253, 1892; Ueber die tierischen Wachstums, Arch. f. Entw. Mech. xv, p. 669, 1902-3; Davenport, On the role of water in growth, Boston Soc. N.H. 1897; Ida H. Hyde in Amer. Journ. Physiology, xii, p. 241, 1905; Bottazzi, Osmotischer Druck und elektrische Leitungsfahigkeit der Fliissigkeiten der Organismen, in Asher-Spiro's Ergebnisse der Physiologic, vn, pp. 160-402, 1908; H. A. Murray in Journ. Gener.

Dynamik des

Physiology, ix, p.

1,

1925; J. Gray, The role of water in the evolution of the and A. N. J. Heyn,

terrestrial vertebrates, Journ. Exper. Biology, vi, pp. 26-31, 1928;

Physiology of cell-elongation, Botan. Review, vi, pp. 515-574, 1940. X Cf. C. A. Berger, Carnegie Inst, of Washington, Contributions

xxvu, 1938.

to

Embryology,

THE RATE OF GROWTH

244

actual increase of protoplasm

we

shall

[CH.

speak presently, but the role

of water in growth deserves a passing word, even in our morphological enquiry.

The lower plants only Uve and grow in abundant moisture; lew when the humidity falls below 85 per cent, of saturation, and the mould-fungi, such as Penicillium, need more fungi continue growing

moisture

still

(Fig. 75).

Their hmit

is

reached a

little

^

Humidity I.

75.

Growth of PeniciUium

in relation to humidity.

Growth of PeniciUium Humidity

below

(at

25° C.) *

90%.

OF TURGESCENCE

Ill]

another.

It

has an upper limit of size vaguely characteristic of the fed well and often

species,

and

diet

grows slowly or

it

if

may

it

may reach it in

dwindle down;

wellnigh what size one pleases. left

245

a year

it

may

;

on stinted

be kept at

Certain full-grown anemones were

untended in war-time, unfed and in water which evaporated

down

to half its bulk;

they shrank

up again when fed and cared

Loeb shewed,

down

to

little

beads,

for.

must the

in certain zoophytes, that not only

be turgescent in order to grow, but that this turgescence only so long as the salt-water in which the

when the

salinity

cells lie

amounts to about

water contains some 3-0 to 3-5 per cent, of

but the salinity

much below

falls

before Tubularia exhibits

a further dilution

its full

cells

possible

does not overstep

3-4 per cent.

salts in the

this normal, to

about

Sea-

open

sea,

2-2 per cent.,

turgescence and maximal growth;

deleterious to the animal.

is

is

a limit reached, in the case of

a certain limit of concentration: Tubularia,

and grew

It is likely

enough

that osmotic conditions control, after this fashion, the distribution

and

local

abundance of many zoophytes.

that in certain fish-eggs

(e.g.

Loeb has

also

shewn*

of Fundulus) an increasing concentration,

leading to a lessening water-content of the egg, retards the rate of

segmentation and at

last arrests

some time longer. The eggs of many

it,

though nuclear division goes on

for

insects absorb water in large quantities,

doubling their weight thereby, and their doing so

;

fail

to develop

and sometimes the egg has a

if

even

drought prevents

thin- walled stalk, or else

a "hydropyle," or other structure by which the water

is

taken inf.

In the frog, according to Bialaszewicz J, the growth of the embryo

membrane depends wholly on absorption varies with the temperature, but the amount

while within the vitelline of water.

The

rate

of water absorbed

is

constant, whether growth be fast or slow.

Moreover, the successive changes of form correspond to definite quantities of water absorbed,

The *

solid residue, as

Pfinger's

much

Davenport Has

of which water also shewn,

is

may

intracellular.

even diminish

Archiv, lv, 1893.

t Cf. V. B. Wigglesworth, hised Physiology, 1939, p. 2. % Beitrage zur Kenntniss d. Wachstumsvorgange bei Amphibienerabryonen, Bull. Acad. Sci. de Cracovie, 1908, p. 783; also A. Drzwina and C. Bohn, De Taction. .dea solutions salines sur les larves des batraciens, ibuL 1906. .

:

THE RATE OF GROWTH

246

[ch.

embryo continues to grow in bulk and especially in the higher animals, the water-content diminishes as growth proceeds and age advances; and loss of water is followed, or accompanied, by retardation and A crab loses water as each phase of growth cessation of growth. draws to an end and the corresponding moult approaches; but it notably, while

and weight.

all

But

the while the

later on,

absorbs water in large quantities as soon as the

Moreover, that water

growth begins*.

been shewn by Davenport for the particularly

by Fehhng

is

frog,

in the case of

lost as

by Potts man.

new

period of

growth goes on has for the chick,

Fehhng's results

and

may

be condensed as follows Age

in

weeks (man)

Percentage of water

6

17

22

24

26

30

35

39

97-5

91-8

92-0

89-9

86-4

83-7

82-9

74-2

I

The following Age

in

weeks

illustrate

(frog)

Davenport's results for the frog:

Ill

OF GAIN OR LOSS OF WATER Distribution of water in a hen's egg

247

THE RATE OF GROWTH

248

distributed in the tissues that water

may

[ch.

Land animals have

be*.

evolved from water animals with Httle change in this respect,

though the constant proportion of water A.

newt

regains

loses moisture it

by no

less

is

variously achieved.

by evaporation with the utmost freedom, and rapid absorption through the skin;

lizard in his scaly coat

is less

hable to the one and

the other, and must drink to replace what water

We

are on the verge of a difficult subject

role of

and

water in the living

other more

;

may

common

capable of lose.

when we speak

and.

we soon

or less unexpected things, that osmotic

neither universal nor yet

while a

of the

growth of the organism,

tissues, in the

in the manifold activities of the cell

it

less

learn,

among

equihbrium

is

The yolk

in the living organism.

maintains a higher osmotic pressure than the white of the egg

—so

and the watery body of a jellyfish, though aot far off osmotic equilibrium, has a somewhat less salinity than the sea-water. In other words, its surface acts to some extent as a semipermeable membrane, and the fluid which causes turgescence long as the egg

of the tissues

is

living;

is less

dense than the sea- water outside!.

In most marine invertebrates, however, the body-fluids constituting the milieu interne are isotonic with the milieu externe,

and

vary in these animals pari passu with the large variations to which sea- water

itself

is

subject.

On

the other hand, the dwellers in

more As to fishes, diiferent kinds shew remarkable differences. Sharks and dogfish have an osmotic pressure in their blood and their body fluids little diiferent

fresh-water, whether invertebrates or fishes, have, naturally, a

concentrated

medium

within than without.

* The vitreous humour is nearly all water, the enamel has next to 'none, the grey matter has some 86 per cent., the bones, say 22 per cent. lung and kidney take up mor^ than they can hold, and so become excretory or regulatory organs. Eggs, whether of dogfish, salmon, frogs, snakes or birds, are composed, roughly speaking, of half water and half solid matter. Chrysaora, Bull. Soc. Biol, t Cf. {int. al.) G. Teissier, Sur la teneur en eau. .de de France, 1926, p. 266. And especially A. V. Hill, R. A. Gortner and others. On the state of water in colloidal and living systems. Trans. Faraday Soc. xxvi, ;

'

.

pp. 678-704, 1930. For recent literature see (e.g.) Homer Smith, in Q. Rev. Biol. vn, p. 1, 1932; E. K. Marshall, Physiol. Rev. xiv, p. 133, 1934; Lovatt Evans, Recent Advances in Physiology, 4th ed., 1930; M. Duval, Recherches. .sur le milieu interieur des animaux aquatiques, Thhe, Paris, 1925; Paul Portier, Physiologic des .

iii, Paris, 1938; G. P. Wells and I. C. Ledingham, Effects of a hypotonic environment, Journ. Exp. Biol, xvii, pp. 337-352, 1940.

animaux marins. Chap,

:

OF OSMOTIC REGULATION

Ill]

from that of the sea-water outside: for

diiferences,

that

instance

diminished, and the molecular concentration

other hand, have a

within

much

are

eked out by large

is

The marine

accumulations of urea in the blood.

249

but with certain chemical chlorides

the

;

teleosts,

on the

lower osmotic pressure within than that

much

and only a httle higher than that of their Some, hke the conger-eel, maintain an all but fresh-water allies. constant internal concentration, very different from that outside; and this fish, like others, is constantly absorbing water from the sea of the sea-water outside,

it

must be exuding or excreting

differ greatly in their

common

Other teleosts

salt continually*.

powers of regulation and of tolerance, the

we may come

stickleback (which

across in a pool or in the

middle of the North Sea) being exceptionally tolerant or "euryhalinef." Physiology becomes "comparative" when it deals with differences such as these,

of just such differences: interieur, qui est

and Claude Bernard foresaw the existence "Chez tous les etres vivants le milieu I'organisme, conserve les rapports

un produit de

necessaires d'echange avec le milieu exterieur;

I'organisme devient plus parfait s'isole

le

mais a mesure que

milieu organique se specifie, et

en quel que sorte de plus en plus au milieu ambiant J." Claude if I mistake not, on Bichat's earUer concept,

Bernard was building,

famous

in its day, of

life

as

"une alternation

habituelle d'action de

la part des corps exterieurs,* et de reaction de la part

vivant":

out of which grew his

still

du corps

more famous aphorism, "La

vie est I'ensemble des fonctions qui resistent a la mort§."

One *

crab, like one fish, differs widely

from another

in its

power

Probably by help of Henle's tubules in the kidney, which structures the dogfish But the gills have their part to play as water-regulators, as

does not possess.

also, for instance, in the crab.

t

The grey mullets go down

brackish or nearly fresh- water.

and Brunelli

to the sea to spawn, but

The several

sets forth, as follows, the

M. auratus saliens chelo capita

cephalus

species differ

may

much

live

and grow

in

in their adaptability,

range of salinity which each can tolerate

24-35 per mille 16-40 10-40 5-40

4^0

For a disX Introduction d Vetude de la medecine experimentale, 1855, p. 110. cussion of this famous concept see J. Barcroft, "La fixite du milieu interieur est la condition §

Sur

de

la vie libre," Biol. Reviews, viii, pp.

la vie et la mart, p. 1.

24-87, 1932.

;

THE RATE OF GROWTH

250

[ch.

of self-regulation; and these physiological differences help to explain, in both cases, the limitation of this species or that to

brackish, or

more or

less saline, waters.

more or

less

In deep-sea crabs {Hyas,

for instance) the osnjotic pressure of the blood keeps nearly to that

of the milieu exteme, and dilution of the latter;

many days

can hve for its

usual sahnity.

near so far;

falls

quickly and dangerously with any

but the httle shore-crab (Cardnus moenas) in sea-water diluted

Meanwhile

its

own

down

to one-quarter of

fluids dilute slowly,

in other words, this crab

but not

combines great powers of

osmotic regulation with a large capacity for tolerating osmotic ,

gradients which are beyond

balance

is

maintained

is

its

power to

regulate.

yet but httle understood.

that certain organs or tissues, especially the

gills

How

the unequal

But we do know and the antennary

gland, absorb, retain or ehminate certain elements, or certain ions, faster

than others, and faster than mere diffusion accounts

for;

in

other words, "ionic*' regulation goes hand in hand with "osmotic" regulation, as a distinct

and even more fundamental phenomenon*. and only natural that quickened

This at least seems generally true respiration





and increased oxygen-consumption accompany

one-sided conditions:

in other words, the "steady state"

all

such

only maintained by the doing of work and the expenditure of energyf. To the dependence of growth on the uptake of water, and to the is

phenomena of osmotic balance and its regulation, HoberJ and also Loeb were inclined to refer the modifications of form which certain phyllopod Crustacea undergo when the highly sahne waters which

they inhabit are further concentrated, or are abnormally diluted. Their

growth

is

retarded

individuals from the

and they become

more

by increased concentration, so that appear stunted and dwarfish

saline waters

altered or transformed in other ways, suggestive

of "degeneration," or a failure to attain * See especially D. A.

full

and perfect develop-

Webb, Ionic regulation in Cardnus moenas, Proc. R.S. (B), cxxix, pp. 107-136, 1940. t In general the fresh- water Crustacea have a larger oxygen -consumption than the marine. Stenohaline and euryhaline are terms applied nowadays to species which are. confined to a narrow range of salinity, or are tolerant of a wide one. An extreme case of toleration, or adaptability, is that of the Chinese woolly-handed crab, Eriockeir, which has not only acclimatised itself in the North Sea but has ascended the Elbe as far as Dresden. X R. Hober, Bedeutung der Theorie der Losungen fiir Physiologic und Medizin, Biol. Centralbl. xix, p. 272, 1899.

OF OSMOTIC REGULATION

Ill]

The consumption more and more

Important physiological changes ensue.

merit*.

251

of oxygen increases greatly in the stronger brines, as active " osmo-regulation "

is

required.

The

rate of multiplication

is

and parthenogenetic reproduction is encouraged. In the less sahne waters male individuals, usually rare, become plentiful, and here the females bring forth their young alive males disappear altogether in the more concentrated brines, and then the females increased,

;

lay eggs, which, however, only begin to develop

when the sahnity

somewhat reduced.

is

The best-known case

is

the

found in one form or another

little

brine-shrimp, Artemia salina,

the world over, and

all

first

discovered

nearly two hundred years ago in the salt-pans at Lymington.

Among many

A. milhausenii, inhabits the natron-

allied forms, one,

Egypt and Arabia, where, under the name of "loul," or "Fezzan-worm," it is eaten by the Arabsf. This fact is interesting,

lakes of

because

indicates (and investigation has apparently confirmed)

it

that the tissues of the creature are not impregnated with the

is

medium

fishes in

in

which

it

hves.

salt,

as

In short Artemia, hke teleostean

the sea, hves constantly in a "hypertonic

medium"; the

body, the milieu interne, are no more salt than are those

fluids of the

any ordinary crustacean or other animal, but contain only some NaCl J while the milieu externe may contain from 3 to 30 per cent, of this and other salts; the skin, or body- wall, of. the creature acts as a "semi-permeable membrane," through which the dissolved salts are not permitted to diffuse, though water passes of

0-8 per cent, of

freely.

grow

When large

,

brought into a lower concentration the animal

and turgescent,

may

until a statical equilibrium, or steady

state, is at length attained.

Among

the structural changes which result from increased con-

* Schmankewitsch, Zeitschr. Schraankewitsch f. wiss. Zool. xxix, p. 429, 1877. has made equally interesting observations on change of size and form in other organisms, after some generations in a milieu of altered density ; e.g. in the flagellate

infusorian Ascinonema acinus Biitschli.

t These "Fezzan- worms," when first described, were supposed to be "insects' eggs"; cf. Humboldt^ Personal Narrative, vi, i, 8, note; Kirby and Spence, Letter x. X See D. J.

N4erland. Zool. Cf.

Kuenen, Notes, systematic and physiological, on Artemia, Arch. iii,

pp. 365-449, 1939;

cf.

also Abonyi, Z.f. w. Z. cxiv, p. 134, 1915.

Mme. Medwedewa, Ueber den osmotischen Druck

in Ztsch. f. vergl. Physiolog. v, pp. 547-554, 1922.

der

Haemolymph

v.

Artemia;

THE RATE OF GROWTH

252

[CH.

centration of the brine (partly during the hfe-time of the individual,

but more markedly during the short season which suffices for the development of three or four, or perhaps more, successive generations),

it

is

found that the

and the

bristles,

tail

these changes corresponding to specific characters of

comes to bear fewer and fewer

themselves tend at last to disappear:

tail-fins

what have been described as the still more extreme

A. milhausenii, and of a

form, A. koppeniana] while on the other hand, progressive dilution of the water tends to precisely opposite conditions, resulting in '

forms which have also been described as separate

u

species,

and even

^wwwWWWH I

I Artewia

Callaonella

s.str.

Brine-shrimps (Artemia), from more or less saline water. Upper figures shew tail-segment and tail-fins; lower figures, relative length of cephalothorax and abdomen. After Abonyi.

Fig. 77.

referred to a separate genus, Callaonella, closely akin to Branchipus is a marked change and hind portions of the body, of the cephalothorax and abdomen: the latter

Pari passu with these changes, there

(Fig. 77).

in the relative lengths of the fore

that

is

to

say,

growing relatively longer, the Salter the water. In other words, not only is the rate of growth of the whole animal lessened by the sahne concentration, but the specific rates of growth in the parts of its

body are

itself to

relatively changed.

numerical statement, and Abonyi has shewn that we

construct a very regular curve, of the creature's

water;

This latter phenomenon lends

abdomen

and the several

by

may

plotting the proportionate length

against the salinity, or density, of the

species of Artemia,

with

all

their other

correlated specific characters, are then found to occupy successive,

more or

less well-defined,

and more or

less

extended, regions of the

OF THE BRINE-SHRIMPS

Ill]

In short, the density of the water

curve (Fig. 78). "specific," that

253

we might

briefly define

is

so clearly

Artemia jelskii, for instance,

as the Artemia of density 1000-1010 (NaCl), or all but fresh water,

and the typical A. salina 1018-1025, and so on*.

(or principalis) as the

Artemia of density Koppeniana

"""^ 160

140

120

100

1000

1020

1040

1060

1080

1100

Density of water Fig. 78,

Percentage ratio of length of abdomen to cephalothorax After Abonyi.

in brine -shrimps, at various salinities.

These Artemiae are capable of living in waters not only of great density, but of very varied chemical composition,

say

how

far

properties

and

it is

they are safeguarded by semi-permeabihty or by

and reactions of the

living colloids "j".

hard to specific

The natron-lakes,

* Different authorities

have recognised from one to twenty species of Artemia. 1910) reduces the salt-water forms to one species with four varieties, but keeps A. jelskii in a separate sub-genus. Kuenen suggests two species, A. salina and gracilis, one for the European and one for the American forms. According to Schmankewitsch every systematic character can be shewn to vary with the external medium. Cf. Professor Labbe on change of characters, specific and even generic, of Copepods according to the ^H of saline waters at

Daday de Dees {Ann.

Le

sci. nat.

March 10, 1928. compare Wo. Ostwald's old experiments on Daphnia, which died in a pure solution of NaCl isotonic with normal sea-water. Their death was not to be explained on osmotic grounds; but was seemingly due to the fact that the organic gels do not retain their normal water- content save in the presence of such concentrations of MgClj (and other salts) as are present in sea-water. t

Croisic, Nature,

We may

THE RATE OF GROWTH

254

for instance, contain large quantities of

magnesium sulphate; and

the Artemiae continue to live equally well in

where

this salt, or

common

[ch.

artificial

solutions

where calcium chloride, has largely replaced the

salt of the

more usual

habitat.

Moreover, such waters as

those of the natron-lakes are subject to great changes of chemical

composition as evaporation and concentration proceed, owing to the

but

different solubilities of the constituent salts;

it

appears that

the forms which the Artemiae assume, and the changes which they

undergo, are identical, or indistinguishable, whichever of the above salts

happen to

the same time

exist or to

we

still

predominate in their saline habitat.

lack, so far as I

experiments which shall

tell

know, the simple but

At

crucial

us whether, in solutions of different

chemical composition,

it is at

trations (that is to say,

under conditions where the osmotic pressure,

equal densities, or at isotonic concen-

and consequently the rate of diffusion, is identical), that the same changes of form and structure are produced and corresponding phases of equihbrium attained. Sea-water has been described as an instance of the "fitness of the

environment*" put to

for the

maintenance of protoplasm in an appropriate

but our Artemias

milieu; it,

makes

shift

suffice to

shew how nature, when hard is wholly abnormal

with an environment which

and anything but "fit." While Hober and others f have referred all these phenomena to osmosis, Abonyi is inclined to believe that the viscosity, or mechanical resistance, of the fluid also reacts upon the organism; and other possible modes of operation have been suggested. But we may take it for certain that the phenomenon as a whole is not a simple one. We should have to look far in organic nature for

what the physicist would is

call

simple osmosis %

;

and assuredly there

always at work, besides the passive phenomena of intermolecular

* L. H. Henderson, The Fitness of the Environment, 1913. t Cf. Schmankewitsch, Z. f. w. Zool. xxv, ISTi); xxix, 1877,. etc.; transl. in appendix to Packard's Monogr. of N. American Phyllopoda, 1^83, pp. 466-514;

Daday de

Dees, Ann. Sci. Nat. (Zool), (9), xi, 1910; Samter und Heymons, Abh. Akad. Wiss. 1902; Bateson, Mat. for the Study of Variation, 1894, pp. 96-101; Anikin, Mitlh. Kais. Univ. Tomsk, xiv: Zool. Centralhl. vi, pp. 756-760, 1908; Abonyi, Z.f. w. Zool. cxiv, pp. 96-168, 1915 (with copious bibliography), etc.

d.

K.

pr.

% Cf. C. F.

J. Duclaux,

A. Pantin,

Body

Chimie apjMquee a

fluids in animals, Biol. Reviews, \i, p. 4,

la biologic, 1937,

ii,

chap.

4.

1931;

OF CATALYTIC ACTION

Ill]

some other

diffusion,

255

activity to play the part of a regulatory

mechanism*.

On growth and In ordinary chemical reactions

catalytic action

we have

to deal (1) with a specific

due

velocity proper to the particular reaction, (2) with variations

to temperature

and other physical conditions,

(3)

with variations

due to the quantities present of the reacting substances, according to Van't Hoff's "Law of Mass Action," and (4) in certain cases with variations due to the presence of "catalysing agents," as BerzeHus called

them a hundred years agof.

law of mass involves a steady

In the simpler reactions, the

slo wing-down of the process as

the

amount of substance diminishes: more or less evaded in the organism,

reaction proceeds and as the initial

a phenomenon, however, which

is

part of whose energies are devoted to the continual bringing-up of

supphes. Catalytic

action

occurs

when some

substance,

often

in

very

and by its presence produces or "a way round," without the by opening accelerates a reaction It diminishes catalysing agent itself being diminished or used up J. the resistance somehow little as we know what resistance means minute quantity,

is

present,



* According to the empirical canon of physiology,

expresses

it

(Arch,

rfc

that,

as

Leon Fredericq maniere que

Zool. 1885), '*L'etre vivant est agence de telle

chaque influence pertyrbatrice provoque d'elle-meme la mise en activite de Tappareil compensateur qui doit neutraliser et reparer le dommage." Herbert Spencer had conceived a similar principle, and thought he recognised in it the vis medicutrix. Nahirae. It is the physiological analogue of the "principle of Le Chatelier " (1888), with this important difference that the latter is a rigorous and quantitative law, The close relation between the two is ba8e ..^pendent

is

that the agreement of cell-forms with the forms which

experiment and mathematical theory assign to liquid

surfaces under the influence of surface-tension

often so typically manifested that

we

is

so frequently

and

are led, or driven, to accept

the surface-tension hypothesis as generally apphcable and as equivalent to a universal law.

exceptions are such as to

The occasional call for

throwing doubt on the hypothesis. a

new element

difficulties or

further enquiry, but

fall

apparent short of

Macallum's researches introduce

of certainty, a "nail in a sure place,"

when they

demonstrate that in certain movements or changes of form which

we should

naturally

attribute

to

weakened surface-tension, a

chemical concentration which would naturally accompany such

weakening actually takes cell

They

place.

a chemical heterogeneity

may

further teach us that in the

exist of a very

certain substances being accumulated here

and absent

marked

kind,

there, within

the narrow bounds of the system. "

Such

localised accumulations

can as yet only be demonstrated in

the case of a very few substances, and of a single one in particular

and these few are substances whose presence does not produce, but whose concentration tends to follow^ a weakening of surface-tension. The physical cause of the localised inequalities of surface-tension

We may assume, if we please, that they are due to the prior accumulation, or local production, of bodies which have this direct effect; though we are by no means limited to this remains unknown.

hypothesis. certainties,

But in spite of some remaining difficulties and unwe have arrived at the conclusion, as regards unicellular

organisms, that not only their general configuration but also their part of the general subject of ionic regulation, which has since become a matter of great physiological importance; cf. [int. al.) D. A. Webb, Ionic regulation in Carcinus mnenns, Pror. R,S. (B), cxxix, pp. 107-130, 1940, and many works

is

quoted therein.

first work on unequal and Donnan's fundamental conception of the Donnan equilibrium {Journ. Chem,. Sdv. xcix, p. 1554, 1911) came just at the same time.

It is

curious and interesting that Macallum's

ionic distribution in the tissues

A NOTE ON ADSORPTION

464 departures

[ch. vi

from symmetry may be correlated with the molecular

forces manifested in their fluid or semi-fluid surfaces.

Looking at the physiological which

side, rather

than at the morphological

more properly our own, we see how very important a system is bound to be, even in respect of its surface-area The order of magnitude of the cells which constitute our

is

cellular

alone.

tissues is such as to give a relation of surface to

anything in

all

volume

far

beyond

the structures or mechanisms devised and fabricated

by man. At this extensive surface, capillary energy, a form of energy scarcely utilised by man, plays a large predominant part in the energetics of the organism.

Even

the warm-blooded animal

not in reality a heat-engine; working as temperatures

its

output of energy

Carnot, to be small.

Nor

electrodynamic one.

It is a

is it

an

is

does at almost constant

it

bound, by the principle of

is

electrostatic machine, nor yet

mechanism

an

which chemical energy

in

turns into surface-energy, and, working hand in hand, the two are

transformed into mechanical energy, by steps which are for the

most part unknown*.

We are

led

on by these considerations to

reflect

on the molecular,

We have already spoken in passing of " monomolecular layers," such as Henri Devaux imagined some thirty years ago, and afterwards obtained f, and rather than the histological, structure of the

cell.

such as Irving Langmuir has lately made his own. of every liquid (provided the form and

The

symmetry

of

free surface

its

molecules

Such a surface no mere limit or simple boundary; it becomes a region of great importance and peculiar activity in certain cases, when, for instance, permit) presents a single layer of oriented molecules.

h

protein molecules of vast complexity are concerned.

a morphological

dynamical

where

or

with a molecular structure of

with energetics of

this alien molecule

through: sufl'er

field

field

may

its

own.

It

its

It

is

then

own, and a

becomes a frontier

be excluded and that other be passed

where some must submit to mere adsorption, and others

chemical change.

membrane,

In a word, we begin to look on a surface-layer

visible or invisible, as a vastly

important thing, a place

and a field of peculiar and potent activity. Lippmann imagined a moteur electrocapillaire, unique in the history of

of delicate operations, *

mechanical invention. Cf. Berthelot, Rev. Sci. Dec. 7, 1913. t CL{int. al.) H. Devaux, La structure moleculaire de la Soc. Bot. de France, lxxv, p. 88, 1928.

cellule vegetale, Bull.

CHAPTER

VII

THE FORMS OF TISSUES OR CELL-AGGREGATES

We

pass from the solitary

cell to cells in

contact with one another

—to

what we may call in the first instance "cell-aggregates," through which we shall be led ultimately to the study of complex tissues.

In this part of our subject, as in the preceding chapters,

we

have to consider the

shall

the case of the soUtary least

we shall probably

find,

but, as in

and we may at

by assuming, that the agency of surface-tension is manifest and important. The effect of this surface-tension

begin

especially will

cell,

effect of various forces;

manifest

itself in surfaces

mininiae areae:

where, as Plateau

we must understand by this expression not an absolute but a relative minimum, an area, that was always is

careful

point out,

to

to say, which approximates to

an absolute minimum as nearly as

the circumstances and material exigencies of the case permit.

There are certain fundamental principles, or fundamental equations, besides those we have already considered, which we shall need in our enquiry;

on (on

p.

for instance, the case

which we

briefly

touched

426) of the angle of contact between the protoplasm

and the axial filament

in a Heliozoan,

we

shall

now

find to be but

a particular case of a general and elementary theorem. in terms of Energy, the general which underlies the theory of surface-tension or capillarity*.

Let us re-state as follows, principle

When

a fluid

is

in contact with another fluid, or with a solid or

with a gas, a portion of the total energy of the system (that, namely,

which we

call surface

surface of contact; specific for

energy)

it is

is

proportional to the area of the

also proportional to a coefficient

each particular pair of substances and

these, save only in so far as it

may

temperature or of electrical charge. condition of

minimum potential

is

which

is

constant for

be modified by changes of

Equihbrium, which

is

the

energy in the system, will accordingly

* See Clerk Maxwell's famous article on "Capillarity" in the ninth edition of the Encyclopedia Britannica, revised by Lord Rayleigh in the tenth edition.

:

THE FORMS OF TISSUES

466 be obtained,

by the utmost

caeteris paribus,

[ch.

possible reduction of

the surfaces in contact.

When we have is

true,

three bodies in contact with one another the

but the case becomes a Httle more complex.

drop of some

same

Suppose a

float on another fluid, J5, while both are Here are three surfaces of contact, that of the drop with the fluid on which it floats, and those of air with the one and other of these two; and the whole surface-energy, E, of the

exposed to

fluid,

A, to

C

air,

system consists

of. three'

parts resident in these three surfaces,

Fig. 151.

or of three specific energies, -S^^, -E'^ic

^bc-

equihbrium, or minimal potential energy,

will

tracting those surfaces

whose

and extending those where

be reached by con-

energy happens to be large

specific

it is

"^^^ condition of

small

—contraction

leading to the

production of a "drop,'f and extension to a spreading "film." Floating on water, turpentine gathers into a drop, olive-oil spreads

out in a film; and these, according to the several specific energies, are the ways by which the and equihbrium attained.

A drop will continue

total energy of the

to exist provided its

system

own two

is

diminished

surface-energies

exceed, per unit area, the specific energy of the water-air surface

around: that

is

to say, provided (Fig. 151)

^AB + ^AC > ^BC But if the one fluid happen to be oil and the other water, then the combined energy per unit-area of the oil-water and the oil-air surfaces together is less than that of the water-air surface

E Hence the

oil-air

>E

4-

E

and oil-water surfaces

surface contracts and disappears, the

and the "drop" gives place surface-area

is

a

the air-water

oil

spreads over the water,

to a "film,"

In both cases the total

minimum under

always provided that no external the situation.

increase,

the circumstances of the case, and force,

such as gravity, complicates

OF FLOATING DROPS

VII]

The surface-energy of which we

are speaking here

in that contractile force, or tension, of

to say*.

467 is

manifested

which we have had so much

In any part of the free water-surface, for instance, one

surface-particle

attracts another surface-particle,

and the multi-

But a

water-particle in

tudinous attractions result in equilibrium. the immediate neighbourhood of the drop

may

be pulled outwards,

Pig. 152.

so to speak,

by another

water-particle, but find

side to furnish the counter-pull;

must

therefore be provided

by tensions existing in the other two if we imagine a single particle placed it will be drawn upon by three different in the three surface-planes and whose

In short,

surfaces of contact.

at the very point of contact, forces,

none on the other

the pull required for equiUbrium

whose directions

lie

* It can easily be proved (by equating the increase of energy stored in an increased surface with the work done in increasing that surface), that the tension measured per unit breadth, T^^, is equal to the energy per unit area, EfO,- Surface-

tensions are very diverse in magnitude, -but

all are positive; Clerk Maxwell conceived the existence of negative surface-tensions, but could not point to any certain instance. When blood-serum meets a solution of common salt, the two

fluids hasten to mix,

long streamers of the one running into the other;

remarkable phenomenon,

this

observed by»Almroth Wright {Proc. R.S. (B), xcii, 1921) and called by him " pseudopodial intertraction," was described by Schoneboom (ibid. (A), ci, 1922) as a case of negative surface-tension. But it is a diffusion-phenomenon rather than a capillary one. first





:

THE FORMS OF TISSUES

468

[ch.

magnitudes are proportional to the

specific tensions characteristic

of the three " interfacial " surfaces.

Now

for three forces acting at

a point to be in equihbrium they must be capable of representation, in

magnitude and

direction,

by the three

in order, in accordance with the

sides of a triangle

taken

theorem of the Triangle of Forces.

we know the form of our drop as it floats on the surface (the point of then by drawing tangents P, R, from mutual contact), we determine the three angles of our triangle, and So, if

(Fig. 152),

know

therefore the relative magnitudes of the three surface-tensions

proportional to

Conversely,

its sides.

acting in the directions P, R,

S

we know the three tensions we know the three

if

(viz. Tab, T^c, T^^)

and know from its three angles the form of the section of the drop. All points round the edge of the drop being under similar conditions, the drop must be circular and its figure

sides of the triangle,

that of a sohd of revolution*.

in

The principle of the triangle of forces is expanded, as follows, an old seventeenth-century theorem, called Lamy's Theorem // three forces acting at a point be in equilibrium, each force

proportional

That

the other two.

to say (in Fig. 152)

is

P R S = sin P R ^-7 = -^ :

:

or

(f)

sm

And from each force

this, in turn, is

:

s,

.



S sm

==-

sm p

s"

derive the equivalent formulae

by which

=

R^

+ S^+ 2RS cos

,

QBE

(the

OBD)

supplement of

are

may be deterequal. The point mined by the intersection of two such circles; and the angle DBO is then the angle,

a, required.

Or we may determine graphically, at two points, the radii of curvature P1P2.

Then,

if s

p-

^^

be the length of the arc between them (which

be determined with along a

'

fair

may

accuracy by rolling the margin of the shell

ruler),

cot a

=

(pi



P2)is.

The following method*, given by Blake, will save actual determination of the radii of curvature.

Measure along a tangent to the curve the distance, AC, at which a certain is made by the curve; and from another point B, measure the distance at which the curve makes an equal offset. Then, calling the offset jLt; the arc AB, s; and AC, BE, respectively x^, x^, we have

small offset, CD,

Pj_

=

a;

2^_

2

^ ,

approximately,

/^

cota =

and

2ixs

Of

all

these methods

specific characters, of a *

by which the mathematical

constants, or

given spiral shell ipay be determined, the

For an example of

this

method, see Blake,

loc. cit. p.

251.

THE EQUIANGULAR SPIRAL

796

only one of which first

much

use has been

made

is

[ch.

that which Moseley

employed, namely, the simple method of determining the

relative

breadths of the whorl at distances separated by some

convenient vectorial angle such as 90°, 180°, or 360°.

Very elaborate measurements of a number of Ammonites have been made by Naumann*, by Grabau, by Sandberger f, and by Miiller,

tion f.

among which we may choose a In the following table

I

Ammonites

couple of cases for considera-

have taken a portion of Grabau's intuslabiatus

Ratio of breadth of Ith of

whorls

OF SHELLS GENERALLY

XI]

797

determinations of the breadth of the whorls in Ammonites (Arcestes) these measurements Grabau gives for every 45° of

intuslabiatus; arc,

but

I

have only

set forth successive whorls

one diameter on both sides of the pole.

measurements

namely the

is

I

ratio

measured along between

alternate

therefore the same ratio as Moseley adopted,

of breadth between contiguous whorls along a

ratio

radius vector.

The

have then added to these observed values the a, as obtained from

corresponding calculated values of the angle

our usual formula.

There

is

considerable irregularity in the ratios derived from these

measurements, but

it will

be seen that this irregularity only implies

a variation of the angle of the spiral between about 85° and 87°;

and the values fluctuate pretty regularly about the mean, which Considering the difficulty of measuring the whorls, is 86° 15'. especially towards the centre,

and

in particular the difficulty of

determining with precise accuracy the position of the pole, clear that in such a case as this

we

the law of the equiangular spiral

is

it

is

are not justified in asserting that

departed from.

Ammonites tornatus

THE EQUIANGULAR SPIRAL

7.98

that of a cone with shghtly curving sides there

:

in which, that

[ch. is

to say,

a sHght acceleration of growth in a transverse as compared

is

with the longitudinal direction.

In a tubular whorls

may

spiral,

whether plane or hehcoid, the consecutive

either be (1) isolated

may

or (2) they

and remote from one another;

precisely meet, so that the outer border of one

and the inner border of the next

just coincide;

or (3) they

may

overlap, the vector plane of each outer whorl cutting that of its

immediate predecessor or predecessors. Looking, as

we have

a cone roUed up*,

it is

done, upon the spiral shell as being essentially plain that, for a given spiral angle, intersection

or non-intersection of the successive whorls will

will its inner it is also

depend upon

the

For the wider the cone, the more

apical angle of the original cone.

border tend to encroach on the preceding whorl.

But

plain that the greater the apical angle of the cone, and the

broader, consequently, the cone

itself,

there be between the total lengths of

the greater difference will

its

inner and outer borders.

And, since the inner and outer borders are describing precisely the same spiral about the pole,

we may

consider the inner border

compared with the outer, and as being always identical with a smaller and earUer part of the latter. If A be the ratio of growth between the outer and the inner curve, then, the outer curve being represented by as being retarded in growth as

r

=

ae^ootcc^

the equation to the inner one will >^e

/= or

r'

* is

To speak

=

aAe^cota^ ae^^-y^^^^"",

of a cone "rolling up," and becoming a nautiloid spiral by doing so, is it easy to see how a cone of

a rough and non-mathematical description; nor

and yet remain a cone. But if (i) the centre of a sphere and its radius keep proportional to the distance the centre has moved, the sphere generates as its envelope a circular cone of which the straight line is the axis; and so, similarly, if (ii) the centre of a sphere move along an equiangular spiral and its radius keep proportional to the arc-distance wide angle could

move along a

roll

up,

straight line

along the spiral back to the pole, the sphere generates as shell-surface, or nautiloid spiral.

its

envelope a self-similar

:

OF SHELLS GENERALLY

XI]

and y may then be inner curve

is

799

called the angle of retardation, to

by

subject

which the

virtue of its slower rate of growth.

Dispensing with mathematical formulae, the several conditions

may

be illustrated as follows

In the diagrams (Fig. 385), OF-^F^F^, etc. represents a radius, on which P^, F^, Pg are the points attained by the outer border of the tubular shell after as

And

Pi', P2', P3' are

border '

or

'

;

many

entire consecutive revolutions.

the points similarly intersected

OFjOF' being always

cutting-down factor.

' '

=

which

A,

Then, obviously,

(

1

by the inner

the ratio of growth,

is )

when OP^ is

than

less

Fig. 385.

OP2' the whorls will be separated by an interspace

=

OF^ than that

OP2' they will be in contact

OF^ is

and

(3)

(a);

when OF^

when

(2) is

greater

there will be a greater or less extent of overlapping,

to say of concealment of the surfaces of the earher

later whorls

(c).

very large, that

And is

as a further case

complete, concealment predecessors.

(4), it is

plain that

by the if

to say if OF-^ be greater, not only than

but also than OP3', OP4',

its

(6),

etc.,

by the

we

A be

OF^

have complete, or all but formed whorl of the whole of

shall

last

This latter condition

is

completely attained in

Nautilus pompilius, and approached, though not quite attained, in

N. umbilicatus] is

of A.

There

(5)

and the

difference

constituted accordingly

"species,"

is

also

between these two forms, or

by a

difference in the value

a final case, not easily distinguishable

THE EQUIANGULAR SPIRAL

800

externally from

where P'

(4),

vector to P, and

on the opposite side of the radius

lies

This final condition

therefore imaginary.

is

[ch.

is

exhibited in Argonauta.

The Hmiting values

of A are easily

ascertained.

In Fig. 386 we have portions of two successive whorls, whose corresponding points on the same radius vector (as R and R) are, therefore, ^^*

277-.

Let r and

the two whorls.

it

follows that

and

Now

at a distance apart corresponding to

'

r'

Then,

if

R'

=

ae^^*^

€2^«0t«, .

TTCota

&

it

mean

does not follow at once and obviously proportional between the breadths of

OB is a mean proportional a corollary which requires to be proved; but the

the adjacent whorls, therefore the whole distance

between proof

is

OA

and OC.

easy..

This

is

:

THE EQUIANGULAR SPIRAL

802

The corresponding values of A are

Constant angle

as follows

Ratio (\) of rates of growth of outer and inner border, such as to produce a spiral with interspaces between the wliorls, the breadth of which intei-spaces is a mean proportional between the breadths of the whorls themselves

(a)

90°

1-00 (imaginary) 0-95 0-89 0-85 0-81 0-76 0-57 0-43 0-32 0-23

89 88 87 86 85 80 75 70 65 60 55 50 45 40 35 30

018 013 0090 0-063

0042 0026 0016

As regards the angle of

/= and

.

retardation, y, in the formula

Ae^^o*«,

/=

or

e(^-^)cota^

in the case r'

it

[ch.

is

=

e(2^-y)cota^

evident that

_

^j.

when y

=

log A

27t,

-

(277

that will

-

y) cot a,

mean

that A

=

1.

In

other words, the outer and inner borders of the tube are identical,

and the tube

When

A

is

is

by one continuous

constituted

a very small fraction, that

is

hne.

to say

when the

rates

of growth of the two borders of the tube are very diverse, then

y in

wfll

—tend that

tend towards infinity

is

to say towards a condition

which the inner border of the tube never grows at

condition

Cyfraea

not infrequently approached in nature.

is is

such a case.

But the nearly

I

This

all.

take

parallel-sided

it

that

cone of

Dentalium, or the widely separated whorls of Lituites, are cases

where A nearly approaches unity

in the

cases

one case, and

is still

large

y being correspondingly small while we can easily find where y is very large, and A is a small fraction, for instance

in the other,

;

in Haliotis, in Calyptraea, or in Gryphaea.

For the purposes of the morphologist, thoji, the main result of this last general investigation is to shew that all the various types of ''open" and "closed" spirals,

all

the various degrees of separation

OF SHELLS GENERALLY

XI]

803

or overlap of the successive whorls, are simply the outward expression of a varying ratio in the rate of growth of the outer as

compared with the inner border of the "tubular shell. The foregoing problem of contact, or intersection, of successive whorls

is

a very simple one in the case of the discoid shell but

a more complex one in the turbinate. contact will evidently take place as

For

when the

compared with the outer whorl

is

in the

discoid shell

retardation of the inner

just 360°,

and the shape of

the whorls need not be considered.

As the angle of retardation diminishes from 360°, the whorls stand further and further apart' in an open coil; as it increases beyond 360°, they overlap more and more and when the angle of retardation is infinite, that is to say when the true inner edge of the whorl does not grow at all, then the shell is said to be completely involute. Of this latter condition we have a striking example in Argonauta, and one a Httle more obscure in Nautilus pompilius. ;

In the turbinate shell the problem of contact

is

twofold, for

v.\,

have to deal with the possibihties of contact on the same side of the axis (which

is

what we have

dealt with in the discoid)

and

also

with the new possibiHty of contact or mtersection on the opposite side;

it

is

this latter case

which

absence of an open umbilicus.

will .determine the presence or

It

is

further obvious that, in the

case of the turbinate, the question of contact or no contact will

depend on the shape of the generating curve simple case where this generating curve eUipse, then contact will be

and

;

may

if

we take the

be considered as an

found to depend on the angle which

the major axis of this elhpse makes with the axis of the

The

shell.

question becomes a compHcated one, and the student will find

it

treated in Blake's paper already referred to.

When

one whorl overlaps another, so that the generating curve

cuts its predecessor (at a distance of

27r)

on the same radius vector,

the locus of intersection will follow^ a spiral fine upon the

whjch

is

called the "suture"

by

conchologists.

It

ensemble of spiral lines in space of which, as

whole it

shell

may

be conceived to be constituted

;

a "contact-spiral," or "spiral of intersection."

is

shell,

one of that

we have seen, and we might In discoid

the call

shells,

such as an Ammonite or a Planorbis, or in Nautilus umbilicatus, there are obviously two such contact-spirals, one on each side of

THE EQUIANGULAR SPIRAL

804 the

shell,

that

to the axis.

but

is

In turbinate

somewhat

is

to say one

rare.

on each

shells

We

have

side of a plane perpendicular

such a condition it

the

shell,

and the other

cone of the umbihcus*;

is

for instance in

spectivum, where the one contact-spiral

Fig. 388.

[ch.

is

visible

also possible,

Solarium per-

on the exterior of

Solarium perspectivum.

lies

internally,

winding round the open

but this second contact-spiral

usually

is

imaginary, or concealed within the whorls of the turbinated

Fig.

390.

pretiosa Fig. 389.

Haliotis tuberculata L.

;

the

;

From

Cooke's Spirals.

or ear shell.

Again, in Haliotis, one of the contact-spirals

Scalaria L.

wentletrap.

the ormer,

shell.

is

non-existent, because

of the extreme obUquity of the plane of the generating curve. *

A

beautiful construction: stupendum Naturae artijicium, Linnaeus.

In

OF SHELLS GENERALLY

XI]

Scalaria pretiosa and in Spirula* there

is

805

no contact-spiral, because

the growth of the generating curve has been too slow in comparison

with the vector rotation of there

is

its

plane.

In Argonauta and in Cypraea

no contact-spiral, because the growth of the generating

curve has been too quick.

Nor, of course,

is

Fig.

any contact-

there

spiral in Patella or in Dentalium, because the angle

shell.

is

too small

Turhinella

392.

Lam.;

a

an

napus chank-

Indian

From Chenu.

Thatcheria mirabilis Angas;

Fig. 391.

from a radiograph by Dr A.

Miiller.

ever to give us a complete revolution of the spire.

Thatcheria

which the outhne of the Up is sharply triangular, instead of being a smooth curve with the result that the apex of the triangle forms a conspicuous "genemirabilis

is

a peculiar and beautiful

shell, in

:

rating spiral", which winds round the shell

than the suture

and

is

more conspicuous

itself.

In the great majority of hehcoid or turbinate

shells

the innermost

* "It [Spirula] is curved so as its roundness is kept, and the Parts do not touch one another": R. Hooke, Posthumoics Works, 1745, p. 284.

;

THE EQUIANGULAR SPIRAL

806

or axial portions

of the whorls

"columella"; and to this

is

[ch.

tend to form a soHd axis or

attached the columellar muscle which

on the one hand withdraws the animal within

its shell,

and on the

other hand provides the controlhng force or trammel, by which (in the gastropod) the growing shell

muscle

now it

is

kept in

course.

its spiral

This

apt to leave a winding groove upon the columella (Fig. 373) and then the muscle is spht into strands or bundles, and then is

leaves parallel grooves with ridges or pleats between,

number

of these folds or pleats

may

vary with the

Volutes, or even with race or locahty.

Thus,

and the

species, as in the

among

the curiosities

of conchology, the chank-shells on the Trincomah coast have four

columellar folds or ridges; but of

all

those from Tranquebar, just north

Adam's Bridge, have only three The various forms of

pods, which

we have

(Fig. 392)*.

straight or spiral shells

among

the Cephalo-

seen to be capable of complete definition

by

the help of elementary mathematics, have received a very com-

phcated descriptive nomenclature from the palaeontologists. instance,

For

the straight cones are spoken of as orthoceracones or

bactriticories,

the loosely coiled forms as gyroceracones or mimo-

ceracones, the

more

closely coiled shells, in

which one whorl overlaps

the other, as nautilicones or ammoniticones, and so forth.

In such

a series of forms the palaeontologist sees undoubted and unquestioned evidence of ancestral descent. Palaeontologyf

:

For instance We read

in Zittel's

" The bactriticone obviously represents the primitive

or primary radical of the

Ammonoidea, and the mimoceracone the

next or secondary radical of this order "

;

while precisely the opposite

drawn by Owen, who supposed that the straight chambered shells of such fossil Cephalopods as Orthoceras had been produced by the gradual unwinding of a coiled nautiloid shell {. The mathematical study of the forms of shells lends no support to these conclusion was

* Cf.

R. Winckworth, Proc. Malacol Soc. xxiii,

p. 345, 1939.

t English edition, 1900, p. 537. The chapter is revised by Professor Alpheus Hyatt, to whom the nomenclature is largely due. For a more copious terminology,

an Acquired Characteristic, 1894, p. 422 seq. Cf. also L. F. Spath, The evolution of the Cephalopoda, Biol. Reviews, viii, pp. 418-462, 1933 see Hyatt, Phytogeny of

.

X Th Cf. also

is

latter conclusion is

Graham

adopted by Willey, Zoological Results, 1902, p. 747. Dana Reports, No. 8, Copenhagen, 1931.

Kerr, on Spirula:

— OF VARIOUS CEPHALOPODS

XI]

or

any

in

which the constant angle of the

stichlihe phylogenetic hypotheses*.

60°, that fact in itself does

one

is

not at

If

807

we have two

shells

spire be respectively 80°

all justify

and

an assertion that the

more- primitive, more ancient, or more "ancestral" than the

other.

Nor,

if

we

find a third in

which the angle happens to be

say that this shell

70°, does that fact entitle us to

is

intermediate

between the other two, in time, or in blood relationship, or in

any other sense whatsoever save only the strictly formal and mathematical one. For it is evident that, though these particular arithmetical constants manifest themselves in visible and recognisable differences of form, yet they are not necessarily more deep-seated or significant than are those which manifest themselves

only in difference of magnitude; scarcely ventures to

of

two

draw conclusions

as to the relative antiquity

organisms on the ground that one happens to be

allied

bigger or

and the student of phylogeny

or longer or shorter, than the other.

less,

At the same time, while.it is obviously unsafe to rest conclusions upon such features as these, unless they be strongly supported and corroborated in other ways for the simple reason that there is unlimited room for coincidence, or separate and independent



attainment of this or that magnitude or numerical ratio the other hand

it is

—yet

on

certain that, in particular cases, the evolution

of a race has actually involved gradual increase or decrease in

some one that

is

or

more numerical

factors,

magnitude

itself

some one

to say increase or decrease in

included

more of the When we do meet with

actual and relative velocities of growth.

or

a clear and unmistakable series of such progressive magnitudes or ratios,

forms,

manifesting themselves in a progressive series of ''aUied"

then we have the phenomenon of "orthogenesis."'

orthogenesis series of

*

is

form (and

also of functional or physiological capacity),

Phylogenetic speculation,

biologist,

Mech.

has had

VII,

p.

104,

For

simply that phenomenon of continuous Unes or

fifty

years ago the chief preoccupation of the

caustic critics. Cf. {int. al.) Rhumbler, in Arch. f. Entw. werden immer auf 1898: " Phylogenetische Speculationen

its

.

.

,

Anklang bei den Fachgenossen rechnen dlirfen, sofern nicht ein anderer Fachgenosse auf demselben Gebiet rait gleicher Kenntniss der Dinge und mit gleicher Scharfainn zufallig zu einer

anderen Theorie

genetischer Schliisse

lasst

sich

gekommen

ist.

.

.

.Die Richtigkeit 'guter' phylo-

im schlimmsten Falle anzweifeln, aber direkt

widerlegen lasst sich in der Regel«iicht."

THE EQUIANGULAR SPIRAL

808

[ch.

which was the foundation of the Theory of Evolution, aUke to

Lamarck and

to

Darwin and Wallace; and which we

see to exist

whatever be our ideas of the "origin of species," or of the nature

and

And

origin of "functional adaptations."

mathematical

(as distinguished

morphology bids orthogenesis in

fair to help

many

mind, the

from the purely physical) study of

us to recognise this

cases where

and, on the other hand, to

eye;

my

to

it is

phenomenon of

not at once patent to the

warn us

in

many

other cases that

even strong and apparently complex resemblances in form capable of arising independently, and

may

may

be

sometimes signify no

more than the equally accidental numerical coincidences which are manifested in identity of length or weight or any other simple magnitudes.

I

have already referred to the fact

great

molluscan from.

that, while in general a very

and remarkable regularity of form shell,

We

have

is

yet that complete regularity

characteristic of the is

apt to be departed

clear cases of such a departure in Pujpa, Clausilia

and various Bulimi, where the curved and narrow

spire

is

not conical, but

its -sides

are

in.

The following measurements of three specimens of

Clausilia

shew

a gradual change in the ratio to one another of successive whorls, or in other

words a marked departure from the logarithmic law: Clausilia lamellosa.

(From Chr. Petersen*.)

OF VARIOUS CEPHALOPODS

XI]

changes in the spiral angle

may be

detected even in ammonites which

But let us suppose that the we shall then get a spiral

present nothing abnormal to the eye.

somewhat

spiral angle increases

rapidly;

with gradually narrowing whorls, which condition Oekotraustes, a subgenus of Ammonites.

If

angle a gradually diminishes, and even

falls

have the

spiral curve

Fig. 393.

opening out, as

An ammonitoid

809

it

is

characteristic of

on the other hand, the

away

to zero,

we

shall

does in Scaphites, Ancyloceras

shell (Macroscaphites) to

shew change of

curvature.

and

Lituites, until the spiral coil is replaced

gentle as to

seem

all

by a

spiral curye so

Lastly, there are a few cases,

but straight.

such as BelleropJion expansus and some Goniatites, where the outer

but the whorls become more more involute. Here it is the of growth between the outer and

spiral does not perceptibly change,

"embracing" or the whole

shell

angle of retardation, the ratio

inner parts of the whorl, which undergoes a gradual change.

In order to understand the relation of a close-coiled shell to straighter congeners, to

an Orthoceras,

it

is

compare

(for

its

example) an Ammonite with

necessary to estimate the length of the right

cone which has, so to speak, been coiled up into the spiral shell. Our problem is, to find the length of a plane equiangular spiral, in

terms of the radius and the constant angle radius vector,

PQ

OQ

a tangent to the curve, PQ, or sec

spiral arc

OP.

a.

Then,

if

OP

be a

a line of reference perpendicular to OP, and a, is

equal in length to the

In other words, the arc measured from the pole

equal to the polar tangent*.

And

is

this is practically obvious: for

* Descartes made this discovery, and records it in a letter to Mersenne, 1638. The equiangular spiral was thus the first transcendental curve to be "rectified."

:

THE EQUIANGULAR SPIRAL

810

PP'/PR'

=

dsjdr

=

sec a,

and therefore

sec a

=

sjr,

[ch.

or the ratio of

arc to radius vector.

Fig. 394.

Accordingly, the ratio of

I,

the total length, to

vector up to which the total length

by a simple a

table of secants

;

is

r,

to be measured,

as follows

the radius is

expressed

OF VARIOUS CEPHALOPODS

XI]

Accordingly,

we

see that

(1),

when the constant angle

spiral is small, the shell (or for that

claw)

and

811 of the

matter the tooth, or horn or

scarcely to be distinguished from a straight cone or cylinder;

is

remains pretty

this

angle, say

much

the case for a considerable increase of

from 0° to 20° or more;

(2) for

a considerably greater

increase of the constant angle, say to 50° or rnore, the shell still

only have the appearance of a gentle curve;

teristic close coils of

represented only

on

either side of

the Nautilus or

diameter

;

88°, is

And

(4)

from about as

rapidity.

circle),

Our

of the close coils of a

six to fifteen times as long as its

we approach an

the spiral vanishes in a

enormous

about six times the length

is

more than three times its own while that of an Ammonite, with a constant angle of, say,

from 85° to diameter.

typically

or rather

radius vector,

its

(3)

Ammonite would be

when the constant angle lies within a few degrees about 80°. The coiled up spiral of a Nautilus,

with a constant angle of about 80°, of

would

the charac-

own

angle of 90° (at which point

the length of the coil increases with

spiral

would soon assume the appearance

Nummuhte, and

the successive increments

of breadth in the successive whorls would become inappreciable to

the eye.

The geometrical form of the

shell involves

properties, of great interest to the

many

other beautiful

mathematician but which

it is

not possible to reduce to such simple expressions as we have been content to use.

For instance, we

may

shall express completely the surface of

obtain an equation which

any

shell, in

or of rectangular coordinates (as has been done

by Blake), sible

or in Hamiltonian vector notation*.

(though of

little

terms of polar

by Moseley and It

is

likewise pos-

interest to the naturalist) to determine the

area of a conchoid al surface or the volume of a conchoidal solid,

and to

find the centre of gravity of either surface or solid f.

Blake has further shewn, with considerable elaboration, how we

And may

deal with the symmetrical distortion due to pressure which fossil shells are often

constitute

by

found to have undergone, and how we

calculation their original undistorted form

may

re-

—a problem

which, were the available methods only a Kttle easier, would be * Cf. H. W. L. Hime's Outlines of Quaternions, 1894, pp. 171-173. Also, for more complete and elaborate treatt See Moseley, op. cit. p. 361 seq. ment, Haton de la Goupilliere, op. cit. 1908, pp. 5-46, 69-204.

THE EQUIANGULAR SPIRAL

812

very helpful to the palaeontologist; it is

[ch

Blake himself has shewn,

for, as

easy to mistake a symmetrically distorted specimen of (for

Ammonite

instance) an

But

genus.

it is

for a

new and

same

distinct species of the

evident that to deal fully with the mathematical

problems contained

in,

or suggested by, the spiral shell, would require

a whole treatise, rather than a single chapter of this elementary book.

Let us then, leaving mathematics

aside,

attempt to summarise, and

perhaps to extend, what has been said about the general

possibilities

of form in this class of organisms.

The univalve

The surface

of

any

shell:

a summary

may

whether discoid or turbinate,

shell,

be

imagined to be generated by the revolution about a fixed axis of a closed curve, which, remaining always geometrically similar to itself,

increases its dimensions continually: and, since the scale of

the figure increases in geometrical progression while the of rotation increases in arithmetical,

and the centre of

angle

similitude

remains fixed, the curve traced in space by corresponding points in the generating curve

is,

in all such cases,

an equiangular spiral.

In

discoid shells, the generating figure revolves in a plane perpendicular

to the axis, as in the Nautilus, the Argonaut

turbinate

shells, it follows

revolution,

and the Ammonite.

In

a skew path with respect to the axis of

and the curve in space generated by any given point makes

a constant angle to the axis of the enveloping cone, and partakes, therefore, of the character of a helix, as well as of a logarithmic spiral.; it

may

be strictly entitled a helico-spiral.

Such turbinate or

spiral shells include the snail, the periwinkle

and

all

the

helico-

common

typical Gastropods.

When

the envelope of the shell

being so

—then

our helico-spiral

—and

seldom far from

is

a right cone

is

a loxodromic curve, and

it is

is

obviously

identical with a projection, parallel with the axis, of the logarithmic spiral

of the base.

As

this spiral cuts all radii at a constant angle, so its orthogonal

projection on the surface intersects

all

generatrices,

and consequently

all

under a constant angle* this being the definition of a loxodromic curve on a surface of revolution. Guido Grandi describes this curve for the first time in a letter to Ceva, printed at the end of his Demonstratio theorematum parallel circles,

Hugenianorum * See

circa.

.

.logarithmicam lineam, 1701

R. C. Archibald, op.

descriptive,

1843) calling

it

cit.

1918.

*.

Ohvier discussed

it

again {Rev. de geom.

a "conical equiangular" or "conical logarithmic"

OF VARIOUS UNIVALVES

XI]

The generating whether

figure

may

813

be taken as any section

of'

the

normal, or otherwise incHned to the axis.

parallel,

commonly assumed to be identical with the mouth of the

shell,

It is

very

shell; in

in other and it is sometimes a plane curve of simple form more numerous cases, it becomes compHcated in form and its boundaries do not he in one plane: but in such cases as these we may replace it by its "trace," on a plane at some

which case

;

definite angle to the direction of growth, for

instance

by

its

form as it appears

through the axis of the helicoid generating curve

is

in a section

The

shell.

of very various shapes.

It is circular in Scalaria or Cyclostoma, in Spirula

;

it

and

may be considered as a segment

of a circle in Natica or in Planorbis.

It is

Conus

and

triangular

in

Thatcheria,

or

rhomboidal in Solarium or Potamides. is

very commonly more or

less elliptical

:

It

the

long axis of the ellipse being parallel to the axis of the shell in Oliva

but perpendicular to oblique to

it

and Cypraea; ;

it is

and

In Nautilus

Haliotis.

approximately a semi-ellipse,

in A^. umbilicatus rather

semi-ellipse, the lo'ng axis lying in

Paul Serret (Th. nouv. cylindroconique'"

also been studied

by

;

.

Section of a spiral

^^^^^^> Triton corrugatus Lam. jrom Woodward.

both cases

easy mathematical expression, save

it ""helice

Fig. 395.

more than a

perpendicular to the axis of the shell*.

spiral.

all

many Trochi and

Lamellaria, Sigaretus halio-

toides (Fig. 396)

and

in

in many well-marked cases,^such

as Stomatella,

pomjpilius

it

Its

when

form

it is

is seldom open to an actual circle or

.des lignes a double courbure, 1860, p. 101) called

Haton de

{int. at.) Tissot,

la Goupilliere calls it a '' conhelice.'" It has Nouv. ann. de mathem. 1852; G. Pirondini,

Mathesis, xix, pp. 153-8, 1899; etc. * In Nautilus, the "hood" has sexes,

and these

somewhat dififerent dimensions in the two upon the shell, that is to say upon its constitutes a somewhat broader ellipse in the

differences are impressed

"generating curve."

The

latter

But

young; with advancing age. Somewhat similar differences in the shells of Ammonites were long ago suspected, by d'Orbigny, to be due to sexual differences. (Cf. Willey, Natural male than

in the female.

in other words, the

this difference is not to be detected in the

form of the generating

Science, vi, p. 411, 1895;

cu'rve perceptibly alters

Zoological Results, 1902, p. 742.)

THE EQUIANGULAR SPIRAL

814

[ch.

but an exception to this rule may be found in certain Ammonites, forming the group " Cordati," where (as Blake points out) ellipse;

the curve

is

very nearly represented by a cardioid, whose equation

=

+

cos

is

r

a

(1

When

6).

the generating curves of successive whorls cut one another,

the hne of

intersection

forms the conspicuous heHco-spiral or

loxodromic curve called the suture by conchologists.

The generating curve may grow slowly factor

or quickly;

its

growth-

very slow in Dentalium or Turritella, very rapid in Nerita,

is

It may contain the axis may be parallel to the axis, as in the may be inclined to the axis, as it is in

or Pileopsis, or Haliotis or the Limpet. in its plane, as in Nautilus

;

majority of Gastropods; or

it it

a very marked degree in Haliotis,

In

fact, in Haliotis the generating

B

A Fig. 396.

A, Lanydlaria perspicua; B, Sigaretus haliotoides. After Woodward.

curve

is

so oblique to the axis of the shell that the latter appears

grow by additions to one margin only (cf. Fig. 362), as in the case of the opercula of Turbo and Nerita referred to on p. 775; and this is what Moseley supposed it to do. The general appearance of the entire shell is determined (apart from the form of its generating curve) by the magnitude of three angles; and these in turn are determined, as has been sufficiently to

explained,

by the

angles are

(1)

turbinate

shells,

ratios of certain velocities of growth.

the constant angle of the equiangular spiral

(a);

These (2) in

the enveloping angle of the cone, or (taking half

that angle) the angle (p) which a tangent to the whorls makes with the axis of the shell; and (3) an angle called the "angle of retarda(y), which expresses the retardation in growth of the inner compared with the outer part of each whorl, and therefore

tion" as

measures the extent to which one whorl overlaps, or the extent to

which

it is

separated from, another.

OF VARIOUS UNIVALVES

XI]

The

spiral angle (a) is

=0°;

taken as

but

very small in a limpet, where

it

it

is

evidently

is

though obscured by the Dentalium

shortness

small, but

still

815

of

a

it is usually-

amount,

significant

of the

tubular

sufficient to give the

In

shell.

appearance

amounts here probably to about 30° to 40°. In Haliotis it is from about 70° to 75°; in Nautilus about 80°; and it lies between 80° and 85° or even more, in the majority of a regular curve;

it

of Gastropods*.

The case of

Fissurella

Here we have, apparently,

curious.

is

a conical shell with no trace of spiral curvature, or (in other words)

with a spiral angle which approximates to 0°;

embryonic

shell (as in

distinctly to be seen.

do with here

is

but in the minute

that of the limpet) a spiral convolution

is

would seem, then, that what we have

^to

It

an unusually large growth-factor in the generating

curve, causing the shell to dilate into a cone of very wide angle,

the apical portion of which has become lost or absorbed, and the

remaining part of which curvature.

is

too short to show clearly

In the closely allied Emarginula, there

well-marked spiral in the embryo, which however

hkewise a

is still

manifested

In both cases

in the curvature of the adult, nearly conical, shell.

we have

to do with a very wide-angled cone,

retardation-factor for

its inner,

its intrinsic

is

and with a high The series is

or posterior, border.

continued, from the apparently simple cone to the complete spiral,

through such forms as Calyptraea.

The angle

a,

as

a constant angle.

we have seen, is not always, nor rigorously, In some Ammonites it may increase with age,

the whorls becoming closer and closer; rapidly and even

fall

out, as in Lituites

and

in

many

in age.

similar forms.

It diminishes

It tends to increase notably in ;

and

it

may

decrease

somewhat,

also,

is

the breadths of successive whorls.

What is sometimes we have

called, as

by

some common

land-shells,

decreases in Succinea.

Directly related to the angle a

*

it

Orthocerata, which are slightly curved in youth but straight

the Pupae and Bulimi

of what

in others

to zero, the coiled shell then straightening

the ratio which subsists between

The following

table gives a few

Leslie, the angle of deflection is the

called the spiral angle (a), or obliquity of the spiral.

angle of deflection

is

6° 17' 41", or the spiral angle

83''

breadths of successive whorls, are doubled at each entire

complement

When

the

42' 19", the radiants, or circuit.

THE EQUIANGULAR SPIRAL

816

[ch.

illustrations of this ratio in particular cases, in addition to those

which we have already studied.

Ratio of breadth of consecutive whorls Obtuse Turbinates and Discoids

Pointed Turbinates ...

1-14

...

1-16

* Turritella terebellata

...

1-18

*Turritella imbricata...

...

1-20

...

1 "22

Conus virgo ... XClymenia laevigata Conus litteratus Conus betulinus XClymenia arietina

Telescopium fuscum Terebra suhulata

...

Cerithium palustre

...

Turritella duplicata

...

Melanopsis terebralis Cerithium nodulosum

...

...

1-25

...

...

1-33

...

...

1*40

...

...

1-43

...

...

1-50

...

1-23

%Goniatites bifer

...

...

1-50

...

1-23

* Helix nemoralis

...

...

1-50

...

1-24

* Solarium perspectivum

...

1*50

1-25

Solarium trochleare ... Solarium magnificum ... *Natica aperta ... ... Euomphalus pentangulatus

1-62

* Turritella car inata

...

...

Terebra crenulata

...

...

1-25 1-25

Terebra maculata (Fig. 397) ' ... *Cerithium lignitarum

1-26

Terebra dimidiata

1-28

Planorbis corneus

...

1-75

2-00 2-00

...

2-00

...

...

1-32

Solaropsis pellis-serpentis

...

2-00

...

...

1*34

Dolium zondtum

...

2-10

...

1-34

XGoniatites carinatus

...

2-50

Trochus niloticus (Fig. 398) ... Mitra episcopalis ...

1-41

*Natica glaucina

...

...

3-00

...

3-00

Fusus antiquus

1

...

4-20 6*00

Cerithium sulcatum Fusus longissimus

*Pleurotomaria conoidea

...

Nautilus pompilius

1-43

...

...

-50

Haliotis excavatus

... ...

...

...

...

6-00

Scalaria pretiosa

...

...

1-56

Fusus

...

...

1-71

Hal^otis parvus Delphinula atrata

PhasiaTvella australis

...

1-80

Haliotis rugoso-plicata

...

9-30

Helicostyla polychroa

...

2-00

Haliotis viridis

...

10-00

colosseus

Those marked

*

from Naumann;

y^,

from

Miiller;

we muSt take

into account the

a from the ratio

shells,

for the short table given

of the breadths of consecutive whorls;

on is

p. 791

is

only applicable to discoid

an angle of 90°.

J^

this formula, I

shells, in

which the angle

Our formula, as mentioned on

becomes

For

the rest from Macalisterf.

in order to determine the spiral angle

In the case of turbinate angle

J

...

_

p.

771,

fi

now

^27rsin/?cota

have worked out the following

table.

f Alex. Macahster, Observations on the mode of growth of discoid and turbinated R.S. xviii, pp. 529-532, 1870; Ann. Mag. N.H. (6), iv, p 160, 1870.

shells, Proc.

Cf. also his

Law

1869, p. 327.

of

Symmetry

as exemplified in animal form, Journ. B. Dublin Soc.

XI]

OF VARIOUS UNIVALVES

t^OiOt^iOCOeCMOiCiOOiMO O5l>?DC0^O5Q0l>COiOO5TtCO

§

00»0'-^^000»0»0»0»0>-0 0i0

r o

O ^

o

»o II

-_

^«-

Ot^r-iO©0»00©000»00 (M CO w ^ c^ eo O -^ -^ 00 CO lO O »0 CO ooi>cou:)kO-r A. 89, 95, 136 Quincke, H. H. 322, 'S31, 359, 418, 447, 502, 579, 659 Quirang, D. P. 203 Rabbit, growth of 1077 Rabelais 5 Rabl, K. 60, 487 Radau, M. 38 Radiolaria 694

117, 187;

skull of

INDEX

1112 Rado, Tibor 382 Rainey, George 10, 653 Ram, horns of 878 Raman, 8ir C. V. 656 Rameau and Sarrus 33

Rammer, W. 165 Ramsden, W. 451 Ramulina 413 Ranella 781 Rankine, W. J. Macquorn 355, 755, 997, 1017 Ransom's waves 294 Ranzio, Silvio 242 Raphides 646 Rashevsky, N. 70, 333 Rasumowsky, V. I. 979 Rauber, A. 490, 604, 633, 970, 979 Ray, John 4 Rayleigh, Lord 26, 55, 69, 72, 304, 380, 398, 418, 449, 469, 502, 510,

659 Read, L.

J.

149

Reaumur, R. A. de

12,

162, 271, 528,

684, 781, 1031

Redwood, growth of 237 Reed, H. S. 260 Regan, C. Tate 423 Regeneration 271 Reichenbach, Hans 123 Reichert, C. B. 290 Reid, E. Way mouth 437 Reighart, J, E. 959 Reindeer beetle 209 Reinecke, J. C. M. 785 Reinke, J. 481, 576 Relay crystals 663 Remak, Robert 341, 456 Reusch, B. 55 Revenstorf 979

Reymond, Du Bois 73

Rhabdammina 870 Rheophax 422 Rhinoceros 807, 874, 905 Rhode, von 194

Rhumbler, L. 293, 301, 419, 506, 704, 807, 852, 863, 871, 892 Riccia 593, 677 Rice, J. 400 Richards, 0. W. 153 Richardson, G. M. 650 Richet, Charles 34 Richtmeyer, F. K. 54 Rideal, E. K. 72, 437 Ridge way. Sir W. 1091 Riemann, B. 609 Ripples 510 Rissoa 565 Rivularia 477 Robb, R. C. 187 Robert, A. 483, 556, 568, 601

Robertson, Muriel 43; R. 162; T. B. 113, 130, 257, 331 Robin, Ch. 1020 Robinson, Wilfred 746 Roederer, J. B. 98 Rohrig, A. 437, 892 , Rollefson, Gunner 179 Rolleston, G. 206 Roosevelt, Th. 961 Rorqual, Sibbald's 69, 173 Rose, Gustav 654 Ross, Ronald 146, 158 Rotalia 365, 792, 865 Rouleaux of blood-corpuscles 439 Roulettes 368

Roux, W.

12, 84,

287, 337, 584, 607,

952 Rudolf, G. de M.

980

Runge, F. F. 659 Runnstrom, J. 438 Rusconi, M. 487 Ruskin, John 661, 932 Russell, E. S. 167 Russow, 110 Rutherford, Lord 286 Rutimeyer, L. 906 Rvder, J. A. 330, 360, 899, 937 Rzasnicki, Adolf 1091 Sachs, J. 60, 63, 115, 222, 344, 933, 974 Sachs's rule 450, 475, 567, 601

Sagrina 422 St Ange, G. J. Martin 243 St Loup, R. 162 St Venant, Barre de 884, 891, 967 SaUsbury, E. J. 217 Salpingoeca 408 Salvia, hairs of 488 Salvinia 601

Sambon, L. W. 944 Samec, M. 656, 669 Samter, M. and Heymons 254 Sandberger, G.

796

Saponin 446 Sapphirina 1055 Sarcode 18 Sasaki, Kichiro

198, 301 Saunders, A. M. Carr 148 Savart, F. 380, 771 Saver, A. B. 68 Scklaria 780, 804, 813 Scale, effect of 23, 673 Scammon, R. E. 100, 108 Scaphites 809 Scapholeberis 506 Scapula, human 1082 Scarus 1062 Schacko, G. 866 SchaefFer, G. 60, 162 Schafer, E. A. Sharpey 264

INDEX

1113

Schaper, A. 168 Schaudinn, F, 75, 456 Scheel, K. 363 Schellbach, K. H. 325, 544 Schewiakoff, W. 316, 701 Schimper, C. F. 915 Schistosoma 943 Schleiden, Matthias 341, 546 Schmaltz, A. 969

Shinryo, M. 265 Ship, speed of 31

Schmankewitz, W. 251, 254 Schmidt, Johannes 169, 191, 238, 656; R. 979; W. J. 312, 647, 835 Schofield, R. K. 986 Schoneboom, C. G. 467 Schonfliess, A. 348 Schultze, F. EQhard 692; Max 17,510 Schiitte, D. J. 234 Schutz, Fr. 445

Simmonds, Katherine

Schwalbe, G. 951 Schwan, Albrecht 688

Shrinkage

717 Silkworm, growth of Silicoflagellata



"

Sea-anemones 184, 244 Searle, H. 745 Seasonal growth 232 Sea-urchins 625, 944; eggs of 353, 602; growth of 229 Sectio aurea, s. divina 923

Sedgwick,

Adam

Sedillot, C. E.

79, 342,

344

985

Sedum 628 Segner, J, A. von

351

Selaginella 638 Selection, natural 770, 840, 936 Selenka, L. 653, 689

Seligmann, G. Selous, F. C.

500 961

Semi-permeable membranes Semon, R. 653, 687 Seneca 2

437

132 Sepia 838 Septa 840; of corals 621 Sequoia 237, 240 Serbetis, C. D. 179 Serpula 865 Serret, Paul 813 Shad, growth of 196 Shapley, Harlow 68, 221 Sharpe, D. 728; Sam 544 Sharpey, William 955 Shearing stress 981, 1017, 1040 Sheep 876 Senility

163

Similitude, principle of

Schwann, Theodor 12, 17, 341, 508, 603, 854 Schwendener, S. 359, 481, 973 Scoresby, William 411 Scorpaena 1063 Scorpioid cyme 767 Scot, Michael 923 Scott, W. B. 1081 Scyromathia 1057

563

Shuleikin, W. 909 Siegwart, J. E. 544 Sigaretus 814

Sinclair,

25, 27 98, 107

Mary E. 363

W. 1049 Siphonogorgia 647 Sitter, W. de 384 Slack, H. J. 510 Slator, A. 153 Slipper limpet 166 Sloan, Tod 993 Sloth 899, 1007 Smeaton, John 29 Smellie, W. 543 Smilodon 271 Smith, Adam 143; D. B. 967; Homer 248; Kenneth 65 Smoke 504 Smoluchowski, M. von 74, 439, 450 Smuts, J. C. 1020 Snow-crystals 411, 599, 695 Soap-bubbles 600 Socrates 695 Soding, H. 281 Sohncke, L. A. 348 Solanum Dulcamara 889 Solarium 804, 813 Solecurtus 827 Solen 828 Solger, B. 979 Sollas, W. J. 675, 692 Sorby, H. C. 646, 1037 Sosman, R. B. 521 Southwell, R. V. 502, 885, 967 Spallanzani, L. 271 Span of arms 189 Spandel, Erich 419 Spangenberg, Fr. 559 Spath, L. F. 806 Spatial complexes 610^ Spek, J. 295, 330 Spencer, Herbert 26, 143, 194, 255 Spermatozoon, path of 434 Sperm -cells 441 Sphacelaria 571 Sphaerechinus 229, -75 Sphagnum 634, 641 Spherocrystals 665 Spherulites 655 Sphodromantis 165 Spicules 547; of sponges 679; holothurians 687; radiolaria 697 Spider's web 386 Sinnett, E.

INDEX

1114

Spindle, nuclear 308 Spira mirabilis 758 Spiral, of Archimedes 752 ; logarithmic

748 Spireme 312 829, 831 Spirillum 433 Spirochaetes 383, 410, 432 Spirographis 848 Spirogyra 16, 171, 371, 378, 401, 578,

Spirifer

677 Spirorbis 848, 865 Spirula 805, 815 Splashes 389, 393 Sponge-spicules 675

Sunflower 258, 913 Surface energy 354 Surirella 434 Sutures of cephalopods Suzuki, Kuro 241 Svedberg, T. 65

Swammerdam, Jan

840

164, 512, 528,

12,

531, 538, 604, 785

Swamy,

S. R. 656 Swezy, Olive 431 Swinnerton, H. H. 831 Sydney, G. J. 77

Sylvester, J. J.

2,

1031

Symmetry, meaning of 357

Spottiswoode, W. 1097 164 Srinavasam, P. S. 656 Stability, elastic 1015 Squilla

208 H. 1053 Standard deviation 124 Starch 665 Stark, J. 263 Starting, E. H. 264 Starvation 189, 265 Staudinger, H. 303 Stay, Benedict 531 Stefanowska, Mile 115, 116 Stegocephalus 1059 Stegosaurus 1006, 1009, 1067 Steinbach, C. 973 Steiner, Jacob 357, 376, 734, 936 Steinmann, G. 665 Stellate cells 547 Stenohaline 250 Stentor 275, 424 Stephenson, John 341 Robert 1012 Stag-beetle

Synapta, egg of 689: spicules of 688 Synge, J. L. 911 Synhelia 514 Szava-Kovatz, J. 217 Szent-Gyorgyi, A. von 457 Szielasko, A. 936

Staigmiiller,

;

Stereochemistry 651 Sternoptyx 1062 Sternum of bird 1013 Stevenson, R. L. 20 Stillman, J. D. 993 Stokes, Sir G. G. 71 Stomach, muscles of 743 Stomata 627 Stomatella 813 Strangeways, T. S. 305 Strasburger, E. 642 Straub, J. 440 Straus-Diirckheim, H. E. 38 Stream-lines 941, 965, 1048 Strehlke, F. 385 Strength of materials 973 Strontium 697 Stuart, Norman 667; T. P. A. Studer, T. 647, 657 Succinea 815 SuflFert, F. 55 Sun-animalcule 427

Tabaka, T. 231 Tadpole, growth of 168, 271, 282 Tait, P. G.

2, 59,

Takahasi, K.

354, 459, 609, 924

500

Taonia 476

Tapetum 641 Tate, John 10 Tate Regan, C.

423

Taylor, G. I. 398, 416; Jeremy 526 Teeth 846; of dolphin 899 ; elephant 901, horse 908 .

George

248 Teixiera, F. G. de 318, 755 Teissier,

Temperature

coefficient

16,

Tenebrio 272 Teodoresco, E. C. 241 Terao, A. 231 Terebra 772, 787, 911 Terni, L. 60 Terquem, Olry 378, 544 Tesch, J. J. 199 Tetracitula 487

586 Tetrahedron 497 Tetrakaidekahedron 734 Tetraspores 632 Textularia 867 Thamnastraea 514 Thatcheria 805, 813 Thayer, J. E. 960 Thecosmiha 512 Theel, H. 678 Theocyrtis 716 Thevenot, Jean 538 Thienemann, F. A. L. 936 Thigh-bone 977 Thigmotaxis 360 Thimann, K. V. 263 Thoma, R. 951 Tetractinellida

1050

221

INDEX Thompson, W. R. 146 Thomson, Allen 956; James 418,

502,

521;

J.

Turgor 26, 355,

Arthur

704;

J.J. 395,447; S. H. 156; Warren Sv 147; C. Wyville 705; WiUiam, see Kelvin Thornton, H. G. 151

Threads 380 Thy one 717 Thyroid gland

186, 265 359 Tilesius, B. 939 Time-element 79 Tintenpilze 393 Tintinnus 408 Tischler, G. 339 Tissot, A. 813, 1033 Titanotherium 1003, 1075 Todd, T. Wingate 98 Tomistoma 1065 Tomkins, R. G. 217, 244 Tomlinson, C. 418, 579, 661, 723 Tomotika, S. 398 Topinard, P. 1055 Topology 609 Tornier, G. 1006 Torpedo 449 Torque of inertia 910 Torre, Father G. M. Delia 358 Torsion 887 Tortoise 172, 517 Townsend, C. H. 172 Trachelophyllum 410 Tradeseantia 388 Transformations, theory of 1032 Traube, M. 457, 649 Trees, growth of 235; height of 29 Trembley, Abraham 27, 274 Treutlein, P. 760 Trianaea, hairs of 387 Triangular numbers 512, 760 Triceratium 511 Trichodina 381 Trichomonas 430 Tridacna 219, 824 Triepel, H. 979 Triolampas 716

Tiegs, 0. VV.

'

Tripocyrtis 721 Triton 813 Trituberculy 900 Troehus 556, 601, 820; egg of 601, 602

Trophon 780 Trout, growth of 176, Truman, A. E. 831 Trypanosomes 430 Tubular structures Tubularia 245, 275 Tur, Jan 83 Turbinella 805 Turbo 771, 775

191, 229, 288, 294

971

1115 243, 245

Turner, Sir William Turritella

1081

742, 771, 814

Turtle 519 Tutton, A. E. H. 38, 696, 731 Tylor, E. 380 Tyndall, J. 504, 661; Tyndali's blue 55 Types, Cuvierian 1094 Typha 631

Uca 206 Uhlenhuth, E.

264

Ultra-violet light

241

Unduloid 369 Unio 558 Univalves 812 Urosalpinx 167 Uvarow, B. P. 219 V'acuole, contractile

409 Valin, H. 649 Vallisneri, Antonio

295

Vaginicola

271

Vallisneria 161 403, 743

Valonia

Van Bambeke, Charles 17, 345 Van Beneden, Eduard 300 Van BischofE, Th. 189 Van der Horst, J. 54, 264 V^andermonde, N. 609 Van Heurck, Henri 434 Van Iterson, G. 553, 743, 85.7 Van Rees 550, 595 Van't Hoff, J. H. 1, 220, 255 Variability 118 Variot, G. 100, 110 Varnish, cracks in 517 Varves 667 Vaucheria 376 Vejdovsky, F. 327 Verhaeren, Emile 1097 145

Verhulst, P. F.

Vermetus 777 Verworn, Max 360, 706, 867 Vesque, Julien 646 Vibrations 509

Vicq d'Azyrj, Felix

9

Vierordt, Hermann 108 Vincent, J. H. 510 Vine-spirals 820 Violet, leaf of 1045 Virchow, R. 341, 380, 456 Virgil 527 Vis reyulsionis

279

264 Vies, Fred 70 Viscose

Vogt, Carl Voice 53

649;

Volkamer, A. W. Voltaire 5, 274

H. 251

538

INDEX

1116 Volterra, Vito 143, 159 Volvox 48, 406, 613 Vom Rath, 0. 315

Von Baer, K. E. 3, Von Buch, Leopold Von Mohl, H. 290, Vorsteher, H. 187 Vortex motion 393; Vorticella 462 Voss, H. 63 Vries, H. de 217

Willem, Victor 541 Wiiley, A. 613, 658, 813, 840

11, 83, 86,

286

785, 845

341

437

rings

Wager, H. W. T.

418 Warner, C. E. 457; Gilbert G. T. 418 Wall, W. S. 30 Wallace, A. Russell Wallis, J. 785

Walton, W.

27,

49, 505;

Robert

143

769

Warren, Ernst 506 Warren's truss 981 Wasp's nest 540 Wasteels, C. E. 924 Watase, S. 602 214, 231

Waterhouse, G. H. 541 Watkin, Emrys 183 Watson, C. M. 146; D. S. M. 960; E. R. 510; G. I. 36 Webb, D. A. 250, 463 Weber, E. H. 418; Max 187 Weidenreich, Fr 434, 981 Weismann, A. 288 Wells, G. P. 248 Wenham, F. H. 49 Went, F. W. 263 Werner, A. G. 27, 506; Fritz 214 Westergaard, H. 98

Westwood,

Wittemore,

J. K, 368 Wodehouse, R. P. 67, 631

Woken, Gertrud 325 544;

Walton and Hammond 269 Warburg, O. 291 Warnecke, P. 189

Waterflea

"Williamson, W. C. 654, 871 Willsia 620 Wilson, E. B. 279, 293, 304, 339, 343, 490, 558, 633, 689 Winckworth, R. 176, 777, 806 Wingfield, C. A. 176 Wings of insects 476, 611 Winsor, C. P. 156 Wislicenus, H. 667 Wissler, Clark 125 Withrow, J. R. 241

163 Wettstein, R, von 1038 Weymouth, F. W. 125, 156, 177

Wolfermann, H.

979

Wolff, C. Freiherr von 341; Julius 976, 979; J. C. F. 3, 79, 285 Wolffia 67 Wood, R. W. 690, 853 Woodcock, ear of 1015 Woodhead, Sims 648, 669

Woodland, W. 687 Woods, R. H. 949 Woodward, H. 840 Worthington, A. M. 389; C. B. 214 Wren, Christopher 785 Wright, Almroth 467; Chauncy 93, 544; Sewell 156; T. Strethill 359 737

Wrinch, Dorothy

Wyman,

Jeffries

531, 538, 976

;

Xylotrupa

212 183

Year-classes

Yeast 152, 363 Young, Thomas 989; W. H. Yule, Sir G.

11, 60, 61, 354, 954,

1032 145

Udny

J. 0.

Weyrauch, W. K. 214 Whelpton, P. K. 147 Wherry, D. 881 Whewell, William 137 White, Gilbert 23; M. J. D. 338 Whitman, C. O. 287, 295, 336, 344 Whitworth, W. A. 757 Whyte, R. G. 241 Wiener, Chr. 74, 757 Wiesner, J. 216 Wigglesworth, V. B. 245, 756 Wilcox, F, W. 263 Wildemann, E. de 483 Wilhelmi, H. 687 Willcox, W. E. F. 149

Morrill

976

Zalophus 264 Zamia 515 Zangger, H. 451 Zaphrentis 623 Zebra 1090 Zeising, A. 915, 931 Zeleny, C. 277, 391

Zeuglodon 1024 Zeuthen, H. G. 762 Ziehen, Ch.

189

P. W. 263, 483 Zinanni, G. 634 Zittel, K. A. von 806 Zoogloea 451

Zimmermann,

Zschokke, F. 979 Zsigmondi, Richard- 68 Zweigelenkige Muskejn 963